Just occasionally, when conducting research for a project, one can stumble – almost accidentally, as it were – upon something which is really important.

During preparatory research for a discussion of “chaos theory” – my shorthand term for the various doomsday scenarios which purport to predict the collapse of Western civilization – I happened to notice that the trajectories of world debt and GDP seem to be following an internal mathematical logic of their own. More specifically, each seems to be subject to exponential progression at remarkably consistent rates.

If this is indeed the case, its implications could be far-reaching, which is the reason for sharing this with you here.

I assume that readers are conversant with exponential progression but, just in case, here’s the gist. Essentially, a numerical series can adopt a “j-curve” or “hockey-stick” shape even though the annual rate of change remains the same. Imagine a $1,000 investment, increasing by 5% per year. In the first year, the extra 5% increases the total by $50, to $1,050. By year 17, however, the total has reached over $2,000, so that year’s increase is more than $100, even though the rate of change is still 5%. Each year, then, the quantity added becomes bigger despite the rate of growth being constant.

The first chart applies a constant rate of progression to total world debt. Let’s be clear about the conventions used in this and subsequent charts. All numbers are expressed in international dollars, converted using the PPP (purchasing power parity) method. In all cases, values are stated in constant 2016 dollars. The blue columns on the chart are historic numbers since 1999, with the 2016 data-point being the end of September, since year-end debt figures are not yet available. The red columns are SEEDS projections out to 2022, and the superimposed black line is an exponential progression at a constant rate of change.

Where debt is concerned, the exponential rate which produces a remarkably close fit is 5.2%. In fact, annual variations from this rate have been strikingly small – and, for each of these charts, a close-up version at the end of this article shows quite how exact the correlation has been.

In 2009, and for wholly understandable reasons, total debt did exceed the 5.2% trend-line, but the very modest deleveraging which occurred in the following year restored the relationship.

Let’s be clear about the implications of this. What this chart is telling us is that global debt is growing by a constant 5.2% annually, and has been doing so for more than fifteen years. Come hell or high water – or their economic equivalents of boom and bust – this progression hardly varies at all. Neither, of course, does inflation enter into this, because these are constant-value numbers.

The suggestion, then, is that global credit expansion is subject to some internal mathematical logic of its own. This, in turn, would help to explain why there has been no retreat from borrowing despite the shock effect of the GFC (global financial crisis) of 2008.

The next chart applies the same methods to GDP (gross domestic product), again in constant (2016) PPP-converted dollars.

Once again, the correlation is very close – had the constant rate curve been followed exactly, GDP would have been $92 trillion (rather than the actual $94tn) in 2010, and $107tn (rather than $111tn) in 2015, but these are very minor divergences.

The big difference this time, though, is that the annual rate of compound expansion is 3.2%, rather than the 5.2% revealed by the progression of debt. (Please note that the first three charts used here have proportionate vertical axes, enabling an undistorted visual appreciation of comparisons).

Strikingly, GDP is growing at a trend rate (3.2%) which is significantly lower than the rate (5.2%) at which debt is increasing. And, since the debt number has been bigger than the GDP number from the outset, the quantity of debt being added each year is bigger than the annual amount of growth.

Moreover, the gap between the annual increments of debt and growth is widening – as, of course, mathematically it must.

This largely explains why the ratio of global debt to global GDP has increased from 163% in 1999 to 221% at the end of September last year. It further suggests that, if the incremental rates of change identified here remain in place – and there seems no reason why they shouldn’t – then the debt ratio will reach 246% by 2020 and 272% by 2025. Neither ratio is impossible – after all, debt to GDP is already higher than 272% in countries such as Japan and the United Kingdom.

Again, a fascinating implication of this finding is that GDP, like debt, rises at a constant rate seemingly dictated by internal mathematical logic. To be sure, GDP got ahead of this rate in the years immediately preceding 2008, but then fell back to trend. Once again, the close-up charts at the end of this discussion show how the trend was first exceeded, and then restored, by the boom-and-bust cycle around the GFC.

This, of course, refers to GDP as the authorities measure it, and regular readers will know my view that recorded GDP has been inflated by the spending of borrowed money. The logic here is that, just as investment relinquishes current in favour of future consumption, the spending of borrowed money, since it does the opposite, is a form of negative investment. In fact, debt is simply one component of futurity, which also embraces, most obviously, pension provision. Just as debt has expanded, pension provision has weakened, because both are components of our financial relationship with futurity.

Therefore, the SEEDS system produces estimates of how much borrowed money is used to inflate current consumption at the expense of the future. This creates estimates of underlying GDP, and this series is the subject of the next chart.

As you can see, there is, once again, a remarkably close fit between actual numbers and a compounding rate of change, which in this instance is 1.8%. Because it excludes the estimated impact of spending borrowed money – of mortgaging futurity, that is – the trend rate of expansion is a lot lower than the rate applicable to GDP as recorded officially.

It is, of course, up to you whether you agree, or not, with my view that, by ramping up debt and using a lot of it to fund consumption, we are boosting today’s consumption at the expense of tomorrow’s. This interpretation seems to me to be reinforced by the opening up of enormous shortfalls in provision for futurity, most visibly in pension deficits.

In short, if pension funding is deficient, the amount available to all of us as pensioners in the future will be smaller than it would otherwise have been. This in turn suggests that we are “pillaging the future” to increase current consumption. Of course, the reason why pension deficits are ballooning, ultimately, is that returns on invested assets have collapsed – and this is a direct consequence of cheap money policies, essentially imposed on central banks by the sheer impossibility of servicing today’s debt mountain at historic (higher) rates of interest.

Wherever you stand on this “mortgaging the future” question, the key point to emerge here is surely that both debt and GDP seem to be subject to rates of change which correlate so closely as to suggest that an internal mathematical dynamic is operating.

The immediate conclusions seem to be that:

World debt is growing at a compound annual rate of 5.2%.

GDP is growing at a compounding rate of 3.2%.

Adjusted for the effects of passing off the spending of borrowed money as “growth”, underlying economic output is growing at a trend rate of 1.8% annually.

All of these numbers exclude two factors which are further undermining prosperity at the individual level. The first of these is population growth, which dilutes per capita shares of economic output. The second is the rising trend in the Energy Cost of Energy (ECoE) – by driving up the cost of household essentials, rising ECoEs act as an expanding “economic rent”, undermining prosperity as this is experienced as “discretionary” (ex-essentials) spending capability.

The findings presented here, of course, are global aggregates, and individual countries’ experiences and prospects vary very significantly around these central trends. India, for instance, could go on growing at current underlying rates for several more decades, whilst the British economy could fall apart within five years.

The general conclusion, however, has to be that, because internal rates of growth are pushing up debt much more rapidly than GDP, there will in due course have to be a reset, most probably through the “soft default” process where the real value of accumulated debt is destroyed by a sharply higher inflation.

What this will not do, however, is reset broader futurity as typified by pension deficits – so we will still need to readdress how we allocate resources between the present and the future.